Tangent, secant, cotangent and cosecant powers

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

For even secant powers, reserve sec²x and use u=tan x. For odd tangent powers, reserving sec x tan x and using u=sec x may help. Cotangent/cosecant families are analogous with their derivative signs.

Step-by-step example

1. Reserve sec²x.

   $$\int\tan^3x\sec^2x\,dx$$

2. Substitute u = tan x.

   $$du=\sec^2x\,dx,\quad I=\int u^3\,du$$

3. Return to x.

   $$I=\frac14\tan^4x+C$$